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Contents
2 Getting to Know Your Data
2.1 Data Objects and Attribute Types . . . . . . . . . . . . . . . . .
2.1.1 What Is an Attribute? . . . . . . . . . . . . . . . . . . . .
2.1.2 Nominal Attributes . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Binary Attributes . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Ordinal Attributes . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Numeric Attributes . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Discrete Versus Continuous Attributes . . . . . . . . . . .
2.2 Basic Statistical Descriptions of Data . . . . . . . . . . . . . . . .
2.2.1 Measuring the Central Tendency: Mean, Median, Mode .
2.2.2 Measuring the Dispersion of Data: Range, Quartiles, Variance, Standard Deviation, and Interquartile Range . . . .
2.2.3 Graphic Displays of Basic Statistical Descriptions of Data
2.3 Data Visualization . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Pixel-Oriented Visualization Techniques . . . . . . . . . .
2.3.2 Geometric Projection Visualization Techniques . . . . . .
2.3.3 Icon-Based Visualization Techniques . . . . . . . . . . . .
2.3.4 Hierarchical Visualization Techniques . . . . . . . . . . .
2.3.5 Visualizing Complex Data and Relations . . . . . . . . . .
2.4 Measuring Data Similarity and Dissimilarity . . . . . . . . . . . .
2.4.1 Data Matrix vs. Dissimilarity Matrix . . . . . . . . . . .
2.4.2 Proximity Measures for Nominal Attributes . . . . . . . .
2.4.3 Proximity Measures for Binary Attributes . . . . . . . . .
2.4.4 Dissimilarity on Numeric Data: Minkowski Distance . . .
2.4.5 Proximity Measures for Ordinal Attributes . . . . . . . .
2.4.6 Dissimilarity for Attributes of Mixed Types . . . . . . . .
2.4.7 Cosine Similarity . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
Chapter 2
Getting to Know Your Data
It’s tempting to jump straight into mining, but first, we need to get the data
ready. This involves having a closer look at attributes and data values. Realworld data are typically noisy, enormous in volume (often several gigabytes or
more), and may originate from a hodgepodge of heterogenous sources. This
chapter is about getting familiar with your data. Knowledge about your data is
useful for data preprocessing (Chapter 3), the first major task of the data mining
process. You will want to know the following: What are the types of attributes
or fields that make up your data? What kind of values does each attribute
have? Which attributes are discrete, and which are continuous-valued? What
do the data look like? How are the values distributed? Are there ways we can
visualize the data to get a better sense of it all? Can we spot any outliers? Can
we measure the similarity of some data objects with respect to others? Gaining
such insight into the data will help with the subsequent analysis.
“So what can we learn about our data that’s helpful in data preprocessing?”
We begin in Section 2.1 by studying the various attribute types. These include nominal attributes, binary attributes, ordinal attributes, and numeric
attributes. Basic statistical descriptions can be used to learn more about each
attribute’s values, as described in Section 2.2. Given a temperature attribute,
for example, we can determine its mean (“average” value), median (“middle”
value), and mode (most common value). These are measures of central
tendency, which give us an idea of the “middle” or center of distribution.
Knowing such basic statistics regarding each attribute makes it easier to fill in
missing values, smooth noisy values, and spot outliers during data preprocessing. Knowledge of the attributes and attribute values can also help in fixing
inconsistencies incurred during data integration. Plotting the measures of central tendency shows us if the data are symmetric or skewed. Quantile plots,
histograms, and scatter plots are other graphic displays of basic statistical descriptions. These can all be useful during data preprocessing, as well as provide
insight into areas for mining.
The field of data visualization provides many additional techniques for viewing data through graphical means. These can help identify relations, trends,
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CHAPTER 2. GETTING TO KNOW YOUR DATA
and biases “hidden” in unstructured data sets. Techniques may be as simple
as scatterplot matrices (where two attributes are mapped onto a 2D grid) to
more sophisticated methods such as tree-map (where a hierarchical partitioning
of the screen is displayed based on the attribute values). Data visualization
techniques are described in Section 2.3.
Finally, we may want to examine how similar (or dissimilar) data objects are.
For example, suppose we have a database where the data objects are patients,
described by their symptoms. We may want to find the similarity or dissimilarity
between individual patients. Such information can allow us to find clusters of
like patients within the dataset. It may also be used to detect outliers in the
data, or to perform nearest-neighbor classification. (Clustering is the topic of
Chapters 10 and 11, while nearest-neighbor classification is discussed in Chapter
8.) There are many measures for assessing similarity and dissimilarity. In
general, such measures are referred to as proximity measures. You can think of
the proximity of two objects as a function of the distance between their attribute
values, although proximity can also be calculated based on probabilities rather
than actual distance. Measures of data proximity are described in Section 2.4.
In summary, by the end of this chapter, you will know the different attribute
types, and basic statistical measures to describe the central tendency and dispersion (spread) of attribute data. You will also know techniques to visualize
attribute distributions, and how to compute the similarity or dissimilarity between objects.
2.1
Data Objects and Attribute Types
Data sets are made up of data objects. A data object represents an entity.
In a sales database, the objects could be customers, store items, or sales, for
instance. In a medical database, the objects may be patients. In a university
database, the objects could be students, professors, and courses. Data objects
are typically described by attributes. Data objects can also be referred to as
samples, examples, instances, data points, or objects. If the data objects are
stored in a database, they are data tuples. That is, the rows of a database
correspond to the data objects, and the columns correspond to the attributes.
In this section, we define attributes and look at the various attribute types.
2.1.1
What Is an Attribute?
An attribute is a data field, representing a characteristic or feature of a data
object. The nouns attribute, dimension, feature, and variable are often used
interchangeably in literature. The term dimension is commonly used in data
warehousing. Machine learning literature tends to use the term feature, while
statisticians prefer the term variable. Data mining and database professionals
commonly use the term attribute. We use the term attribute here as well. Attributes describing a customer object can include, for example, customer ID,
name, and address. Observed values for a given attribute are known as obser-
2.1. DATA OBJECTS AND ATTRIBUTE TYPES
5
vations. A set of attributes used to describe a given object is called an attribute
vector (or feature vector ). The distribution of data involving one attribute (or
variable) is called univariate. A bivariate distribution involves two attributes,
and so on.
The type of an attribute is determined by the set of possible values the
attribute can have. Attributes can be nominal, binary, ordinal, or numeric. In
the following subsections, we introduce each type.
2.1.2
Nominal Attributes
Nominal means “relating to names.” The values of a nominal attribute are
symbols or names of things. Each value represents some kind of category, code,
or state and so nominal attributes are also referred to as categorical. The
values do not have any meaningful order about them. In computer science, the
values are also known as enumerations.
Example 2.1 Nominal attributes. Suppose that Hair color and Marital status are two
attributes describing person objects. In our application, possible values for
Hair color are black, brown blond, red, auburn, grey, and white. Marital status
can take on the values single, married, divorced, and widowed. Both Hair color
and Marital status are nominal attributes. Occupation is another example, with
the values teacher, dentist, programmer, farmer, and so on.
Although we said that the values of a nominal attribute are symbols or
“names of things”, it is possible to represent such symbols or “names” with
numbers. With Hair color, for instance, we can assign a code of 0 for black, 1
for brown, and so on. Customor ID is another example of a nominal attribute
whose possible values are all numeric. However, in such cases, the numbers are
not intended to be used quantitatively. That is, mathematical operations on
values of nominal attributes are not meaningful. It makes no sense to subtract
one customer ID number from another, unlike, say, subtracting an age value
from another. (Age is a numeric attribute). Even though a nominal attribute
may have integers as values, it is not considered a numeric attribute because the
integers are not meant to be used quantitatively. We will say more on numeric
attributes in Section 2.1.5.
Because nominal attribute values do not have any meaningful order about
them and are not quantitative, it makes no sense to find the mean (average)
value or median (middle) value for such an attribute, given a set of objects. One
thing that is of interest, however, is the attribute’s most commonly occurring
value. This value, known as the mode, is one of the measures of central tendency.
We will say more about how to compute the measures of central tendency in
Section 2.2.
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2.1.3
CHAPTER 2. GETTING TO KNOW YOUR DATA
Binary Attributes
A binary attribute is a nominal attribute with only two categories or states:
0 or 1, where 0 typically means that the attribute is absent, and 1 means that
it is present. Binary attributes are referred to as Boolean if the two states
correspond to true and false.
Example 2.2 Binary attributes. Given the attribute Smoker describing a patient object,
1 indicates that the patient smokes, while 0 indicates that the patient does not.
Similarly, suppose the patient undergoes a medical test that has two possible
outcomes. The attribute Medical test is binary, where a value of 1 means the
result of the test for the patient is positive, while 0 means the result is negative.
A binary attribute is symmetric if both of its states are equally valuable and
carry the same weight; that is, there is no preference on which outcome should
be coded as 0 or 1. One such example could be the attribute gender having the
states male and female. A binary attribute is asymmetric if the outcomes of
the states are not equally important, such as the positive and negative outcomes
of a medical test for HIV. By convention, we shall code the most important
outcome, which is usually the rarest one, by 1 (e.g., HIV positive) and the other
by 0 (e.g., HIV negative).
2.1.4
Ordinal Attributes
An ordinal attribute is an attribute whose possible values have a meaningful
order or ranking among them, but the magnitude between successive values is
not known.
Example 2.3 Ordinal attributes. Suppose that Drink size corresponds to the size of drinks
available at a fast food restaurant. This nominal attribute has three possible
values – small, medium, and large. The values have a meaningful sequence
(which corresponds to increasing drink size), however, we cannot tell from the
values how much bigger, say, a medium is from a large. Other examples of
ordinal attributes include Grade (e.g., A+, A, A−, B+, and so on) and Professional rank. Professional ranks can be enumerated in a sequential order, such
as assistant, associate, and full for professors, and private, private first class,
specialist, corporal, sergeant for army ranks.
Ordinal attributes are useful for registering subjective assessments of qualities that cannot be measured objectively. Hence, ordinal attributes are often
used in surveys for ratings. In one survey, participants were asked to rate how
satisfied they were as customers. Customer satisfaction had the following ordinal categories: 0: very dissatisfied, 1: somewhat dissatisfied, 2: neutral, 3:
satisfied, and 4: very satisfied.
Ordinal attributes may also be obtained from the discretization of numeric
quantities by splitting the value range into a finite number of ordered categories
2.1. DATA OBJECTS AND ATTRIBUTE TYPES
7
as described in Chapter 3 on data reduction.
The central tendency of an ordinal attribute can be represented by its mode
and its median (the middle value in an ordered sequence), but the mean cannot
be defined.
Note that nominal, binary, and ordinal attributes are qualitative. That is,
they describe a feature of an object, without giving an actual size or quantity.
The values of such qualitative attributes are typically words representing categories. If integers are used, they represent computer codes for the categories, as
opposed to measurable quantities (e.g., 0 for small drink size, 1 for medium, and
2 for large). In the following subsection we look at numeric attributes, which
provide quantitative measurements of an object.
2.1.5
Numeric Attributes
A numeric attribute is quantitative, that is, it is a measurable quantity, represented in integer or real values. Numeric attributes can be interval-scaled or
ratio-scaled.
Interval-Scaled Attributes
Interval-scaled attributes are measured on a scale of equal-sized units. The
values of interval-scaled attributes have order and can be positive, 0, or negative.
Thus, in addition to providing a ranking of values, such attributes allow us to
compare and quantify the difference between values.
Example 2.4 Interval-scaled attributes. Temperature is an interval-scaled attribute. Suppose that we have the outdoor temperature value for a number of different days,
where each day is an object. By ordering the values, we obtain a ranking of the
objects with respect to temperature. In addition, we can quantify the difference
between values. For example, a temperature of 20◦ C is 5 degrees higher than
a temperature of 15◦ C. Calendar dates are another example. For instance, the
years 2002 and 2010 are 8 years apart.
Temperatures in Celsius and Fahrenheit do not have a true zero-point, that
is, neither 0◦ C nor 0◦ F indicates “no temperature.” (On the Celsius scale, for
example, the unit of measurement is 1/100 of the difference between the melting
temperature and the boiling temperature of water in atmospheric pressure.)
Although we can compute the difference between temperature values, we cannot
talk of one temperature value as being a multiple of another. Without a true
zero, we cannot say, for instance, that 10◦ C is twice as warm as 5◦ C. That
is, we cannot speak of the values in terms of ratios. Similarly, there is no true
zero-point for calendar dates. (The year 0 does not correspond to the beginning
of time.) This brings us to the next type of attribute – ratio-scaled attributes,
for which a true zero-point exits.
Because interval-scaled attributes are numeric, we can compute their mean
value, in addition to the median and mode measures of central tendency.
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CHAPTER 2. GETTING TO KNOW YOUR DATA
Ratio-Scaled Attributes
A ratio-scaled attribute is a numeric attribute with an inherent zero-point.
That is, if a measurement is ratio-scaled, we can speak of a value as being a
multiple (or ratio) of another value. In addition, the values are ordered, and we
can also compute the difference between values. The mean, median, and mode
can be computed as well.
Example 2.5 Ratio-scaled attributes. Unlike temperatures in Celsius and Fahrenheit, the
Kelvin (K) temperature scale has what is considered a true zero-point (0 degrees
K = −273.15◦C): It is the point at which the particles that comprise matter have
zero kinetic energy. Other examples of ratio-scaled attributes include Count
attributes such as Years of experience (where the objects are employees, for
example) and Number of words (where the objects are documents). Additional
examples include attributes to measure weight, height, latitude and longitude
coordinates (e.g., when clustering houses), and monetary quantities (e.g., you
are 100 times richer with $100 than with $1).
2.1.6
Discrete Versus Continuous Attributes
In our presentation, we have organized attributes into nominal, binary, ordinal,
and numeric types. There are many ways to organize attribute types. The types
are not mutually exclusive.
Classification algorithms developed from the field of machine learning often
talk of attributes as being either discrete or continuous. Each type may be
processed differently. A discrete attribute has a finite or countably infinite
set of values, which may or may not be represented as integers. The attributes
Hair color, Smoker, Medical test, and Drink size each have a finite number of
values, and so are discrete. Note that discrete attributes may have numeric
values, such as 0 and 1 for binary attributes, or, the values 0 to 110 for the
attribute Age. An attribute is countably infinite if the set of possible values is
infinite, but the values can be put in a one-to-one correspondence with natural
numbers. For example, the attribute customer ID is countably infinite. The
number of customers can grow to infinity, but in reality, the actual set of values
is countable (where the values can be put in one-to-one correspondence with
the set of integers). Zip codes are another example.
If an attribute is not discrete, it is continuous. The terms “numeric attribute” and “continuous attribute” are often used interchangeably in the literature. (This can be confusing because, in the classical sense, continuous values
are real numbers, whereas numeric values can be either integers or real numbers.) In practice, real values are represented using a finite number of digits.
Continuous attributes are typically represented as floating-point variables.
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
2.2
9
Basic Statistical Descriptions of Data
For data preprocessing to be successful, it is essential to have an overall picture
of your data. Basic statistical descriptions can be used to identify properties of
the data and highlight which data values should be treated as noise or outliers.
This section discusses three areas of basic statistical descriptions. We start
with measures of central tendency (Section 2.2.1), which measure the location
of the middle or center of a data distribution. Intuitively speaking, given an
attribute, where do most of its values fall? In particular, we discuss the mean,
median, mode, and midrange.
In addition to assessing the central tendency of our data set, we also would
like to have an idea of the dispersion of the data. That is, how are the data
spread out? The most common measures of data dispersion are the range,
quartiles, and interquartile range, the five-number summary and boxplots, and
the variance and standard deviation of the data. These measures are useful for
identifying outliers. These are described in Section 2.2.2.
Finally, we can use many graphic displays of basic statistical descriptions
to visually inspect our data (Section 2.2.3). Most statistical or graphical data
presentation software packages include bar charts, pie charts, and line graphs.
Other popular displays of data summaries and distributions include quantile
plots, quantile-quantile plots, histograms, and scatter plots.
2.2.1
Measuring the Central Tendency: Mean, Median,
Mode
In this section, we look at various ways to measure the central tendency of
data. Suppose that we have some attribute X, like salary, which has been
recorded for a set of objects. Let x1 , x2 , . . . , xN be the set of N observed values
or observations for X. These values may also be referred to as “the data set”
(for X) in the remainder of this section. If we were to plot the observations for
salary, where would most of the values fall? This gives us an idea of the central
tendency of the data. Measures of central tendency include the mean, median,
mode, and midrange.
The most common and effective numeric measure of the “center” of a set
of data is the (arithmetic) mean. Let x1 , x2 , . . . , xN be a set of N values or
observations, such as for some numeric attribute X, like salary. The mean of
this set of values is
N
X
xi
x1 + x2 + · · · + xN
=
.
(2.1)
N
N
This corresponds to the built-in aggregate function, average (avg() in SQL),
provided in relational database systems.
x̄ =
i=1
Example 2.6 Mean. Suppose we have the following values for salary (in thousands of
dollars), shown in increasing order: 30, 31, 47, 50, 52, 52, 56, 60, 63, 70, 70,
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CHAPTER 2. GETTING TO KNOW YOUR DATA
110. Using Equation (2.1), we have
x̄
=
=
30 + 36 + 47 + 50 + 52 + 52 + 56 + 60 + 63 + 70 + 70 + 110
12
696
= 58.
12
Thus, the mean salary is $58K.
Sometimes, each value xi in a set may be associated with a weight wi , for
i = 1, . . . , N . The weights reflect the significance, importance, or occurrence
frequency attached to their respective values. In this case, we can compute
x̄ =
N
X
wi xi
i=1
N
X
=
wi
w1 x1 + w2 x2 + · · · + wN xN
.
w1 + w2 + · · · + wN
(2.2)
i=1
This is called the weighted arithmetic mean or the weighted average.
Although the mean is the single most useful quantity for describing a data
set, it is not always the best way of measuring the center of the data. A major
problem with the mean is its sensitivity to extreme (e.g., outlier) values. Even
a small number of extreme values can corrupt the mean. For example, the mean
salary at a company may be substantially pushed up by that of a few highly
paid managers. Similarly, the mean score of a class in an exam could be pulled
down quite a bit by a few very low scores. To offset the effect caused by a small
number of extreme values, we can instead use the trimmed mean, which is
the mean obtained after chopping off values at the high and low extremes. For
example, we can sort the values observed for salary and remove the top and
bottom 2% before computing the mean. We should avoid trimming too large
a portion (such as 20%) at both ends as this can result in the loss of valuable
information.
For skewed (asymmetric) data, a better measure of the center of data is the
median. The median is the middle value in a set of ordered data values. It is
the value that separates the higher half of a data set from the lower half.
In probability and statistics, the median generally applies to numeric data,
however, we may extend the concept to ordinal data. Suppose that a given data
set of N values for an attribute X is sorted in increasing order. If N is odd,
then the median is the middle value of the ordered set. If N is even then the
median is not unique; it is the two middlemost values and any value in between.
If X is a numeric attribute in this case, by convention, the median is taken as
the average of the two middlemost values.
Example 2.7 Median. Let’s find the median of the data from Example 2.2.1. The data
are already sorted in increasing order. There is an even number of observations
(i.e., 12), therefore, the median is not unique. It can be any value within the
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
11
two middlemost values of 52 and 56 (that is, within the 5th and 6th values in
the list). By convention, we assign the average of the two middlemost values as
the median. That is, 52+56
= 108
2
2 = 54. Thus, the median is $54K.
Suppose that we had only the first eleven values in the list. Given an odd
number of values, the median is the middlemost value. This is the 5th value in
this list, which has a value of $52K.
The median is expensive to compute when we have a large number of observations. For numeric attributes, however, we can easily approximate the value.
Assume that data are grouped in intervals according to their xi data values and
that the frequency (i.e., number of data values) of each interval is known. For
example, employees may be grouped according to their annual salary in intervals
such as 10–20K, 20–30K, and so on. Let the interval that contains the median
frequency be the median interval. We can approximate the median of the entire
data set (e.g., the median salary) by interpolation using the formula:
P
N/2 − ( freq)l
median = L1 +
width,
(2.3)
freqmedian
where L1 is the lower boundary
P of the median interval, N is the number of
values in the entire data set, ( f req)l is the sum of the frequencies of all of the
intervals that are lower than the median interval, f reqmedian is the frequency
of the median interval, and width is the width of the median interval.
The mode is another measure of central tendency. The mode for a set of
data is the value that occurs most frequently in the set. Therefore, it can be
determined for qualitative and quantitative attributes. It is possible for the
greatest frequency to correspond to several different values, which results in
more than one mode. Data sets with one, two, or three modes are respectively
called unimodal, bimodal, and trimodal. In general, a data set with two or
more modes is multimodal. At the other extreme, if each data value occurs
only once, then there is no mode.
Example 2.8 Mode. The data of Example 2.2.1 are bimodal. The two modes are $52K and
$70K.
For unimodal numeric data that are moderately skewed (asymmetrical), we
have the following empirical relation:
mean − mode ≈ 3 × (mean − median).
(2.4)
This implies that the mode for unimodal frequency curves that are moderately
skewed can easily be approximated if the mean and median values are known.
The midrange can also be used to assess the central tendency of a numeric
data set. It is the average of the largest and smallest values in the set. This
measure is easy to compute using the SQL aggregate functions, max() and min().
Example 2.9 Midrange. The midrange of the data of Example 2.2.1 is
30+110
2
= $70K.
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CHAPTER 2. GETTING TO KNOW YOUR DATA
Mean
Median
Mode
Mode Mean
Mean
Median
(a) symmetric data
(b) positively skewed data
Mode
Median
(c) negatively skewed data
Figure 2.1: Mean, median, and mode of symmetric versus positively and negatively skewed data. [TO EDITOR NOTE: Suggest using, say, dashed line for
mean, dotted line for median, and dashed-dotted line for mode in each of the
above graphs to improve readability.]
In a unimodal frequency curve with perfect symmetric data distribution,
the mean, median, and mode are all at the same center value, as shown in
Figure 2.1(a).
Data in most real applications are not symmetric. They may instead be
either positively skewed, where the mode occurs at a value that is smaller than
the median (Figure 2.1(b)), or negatively skewed, where the mode occurs at
a value greater than the median (Figure 2.1(c)).
2.2.2
Measuring the Dispersion of Data: Range, Quartiles, Variance, Standard Deviation, and Interquartile Range
We now look at measures to assess the dispersion or spread of numeric data. The
measures include range, quantiles, quartiles, percentiles, and the interquartile
range. The five-number summary, which can be displayed as a boxplot, is useful
in identifying outliers. Variance and standard deviation also indicate the spread
of a data distribution.
Range, Quartiles, and the Interquartile Range (IQR)
To start off, let’s study the range, quantiles, quartiles, percentiles, and the interquartile range as measures of data dispersion.
Let x1 , x2 , . . . , xN be a set of observations for some numeric attribute, X.
The range of the set is the difference between the largest (max()) and smallest
(min()) values.
Suppose that the data for attribute X are sorted in increasing numeric order.
Imagine that we can pick certain data points so as to split the data distribution
into equal-sized consecutive sets, as in Figure 2.2. These data points are called
quantiles. Quantiles are points taken at regular intervals of a data distribution,
dividing it into essentially equal-sized consecutive sets. (We say “essentially”
because there may not be data values of X that divide the data into exactly
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
13
Here will be a graph: not drawn yet. Any volunteer to draw such a
graph?! I have a nice drawing software called SmarDraw. –JH
Figure 2.2: A plot of the data distribution for some attribute X. Here, the
quantiles plotted are quartiles. The three quartiles divide the distribution into
four equal-sized consecutive subsets. The second quartile corresponds to the
median.
equal-sized subsets. For readability, we will refer to them hereafter as equal.)
The k th q-quantile for a given data distribution is the value x such that at most
k/q of the data values are less than x and at most (q − k)/q of the data values
are more than x, where k is an integer such that 0 < k < q. There are q − 1
q-quantiles.
The 2-quantile is the data point dividing the lower and upper halves of the
data distribution. It corresponds to the median. The 4-quantiles are the three
data points that split the data distribution into four equal parts so that each
part represents one fourth of the data distribution. They are more commonly
referred to as quartiles. The 100-quantiles are more commonly referred to as
percentiles; they divide the data distribution into 100 equal-sized consecutive
sets. The median, quartiles, and percentiles are the most widely used forms of
quantiles.
The quartiles give an indication of the center, spread, and shape of a distribution. The first quartile, denoted by Q1 , is the 25th percentile. It cuts off
the lowest 25% of the data. The third quartile, denoted by Q3 , is the 75th
percentile. It cuts off the lowest 75% (or highest 25%) of the data. The second
quartile is the 50th percentile. As the median, it gives the center of the data
distribution.
The distance between the first and third quartiles is a simple measure of
spread that gives the range covered by the middle half of the data. This distance
is called the interquartile range (IQR) and is defined as
IQR = Q3 − Q1 .
(2.5)
Example 2.10 Interquartile range. The quartiles are the three values that split the sorted
data set into four equal parts. The data of Example 2.2.1 contain 12 observations, already sorted in increasing order. Thus, the quartiles for this data are
the 3rd, 6th, and 9th values, respectively, in the sorted list. Therefore, Q1 =
$47K and Q3 is $63K. Thus, the interquartile range is IQR = 63 − 47 = $16K.
(Note that the 6th value is a median, $52K, although this data set has two
medians since the number of data values is even.)
Five-Number Summary, Boxplots, and Outliers
No single numeric measure of spread, such as IQR, is very useful for describing
skewed distributions. Have a look at the symmetric and skewed data distributions
14
CHAPTER 2. GETTING TO KNOW YOUR DATA
of Figure 2.1. In the symmetric distribution, the median (and other measures of
central tendency) splits the data into equal-size halves. This does not occur for
skewed distributions. Therefore, it is more informative to also provide the two
quartiles Q1 and Q3 , along with the median. A common rule of thumb for identifying suspected outliers is to single out values falling at least 1.5 × IQR above
the third quartile or below the first quartile.
Because Q1 , the median, and Q3 together contain no information about the
endpoints (e.g., tails) of the data, a fuller summary of the shape of a distribution
can be obtained by providing the lowest and highest data values as well. This
is known as the five-number summary. The five-number summary of a distribution consists of the median, the quartiles Q1 and Q3 , and the smallest and
largest individual observations, written in the order of M inimum, Q1 , M edian,
Q3 , M aximum.
Boxplots are a popular way of visualizing a distribution. A boxplot incorporates the five-number summary as follows:
• Typically, the ends of the box are at the quartiles, so that the box length
is the interquartile range, IQR.
• The median is marked by a line within the box.
• Two lines (called whiskers) outside the box extend to the smallest (Minimum) and largest (Maximum) observations.
When dealing with a moderate number of observations, it is worthwhile to
plot potential outliers individually. To do this in a boxplot, the whiskers are
extended to the extreme low and high observations only if these values are less
than 1.5 × IQR beyond the quartiles. Otherwise, the whiskers terminate at
the most extreme observations occurring within 1.5 × IQR of the quartiles. The
remaining cases are plotted individually. Boxplots can be used in the comparisons
of several sets of compatible data.
Example 2.11 Boxplot. Figure 2.3 shows boxplots for unit price data for items sold at four
branches of AllElectronics during a given time period. For branch 1, we see that
the median price of items sold is $80, Q1 is $60, Q3 is $100. Notice that two
outlying observations for this branch were plotted individually, as their values of
175 and 202 are more than 1.5 times the IQR here of 40.
Boxplots can be computed in O(nlogn) time. Approximate boxplots can
be computed in linear or sublinear time depending on the quality guarantee
required.
Variance and Standard Deviation
Variance and standard deviation are measures of data dispersion. They indicate
how spread out a data distribution is. A low standard deviation means that
the data observations tend to be very close to the mean, while high standard
deviation indicates that the data are spread out over a large range of values.
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
15
200
180
160
Unit price ($)
140
120
100
80
60
40
20
Branch 1
Branch 2
Branch 3
Branch 4
Figure 2.3: Boxplot for the unit price data for items sold at four branches of
AllElectronics during a given time period.
The variance of N observations, x1 , x2 , . . . , xN , for a numeric attribute X
is
σ2 =
N
N
1 X
1 X 2
(xi − x̄)2 = (
x ) − x̄2 ,
N i=1
N i=1 i
(2.6)
where x̄ is the mean value of the observations, as defined in Equation (2.1). The
standard deviation, σ, of the observations is the square root of the variance,
σ2 .
Example 2.12 Variance and standard deviation. In Example 2.2.1, we found x̄ = $58K
using Equation (2.1) for the mean. To determine the variance and standard
deviation of the data from that example, we set N = 12 and use Equation (2.6)
to obtain
σ2
σ
1
(302 + 362 + 472 . . . + 1102) − 582
12
≈ 379.17
√
≈
379.17 ≈ 19.47.
=
The basic properties of the standard deviation, σ, as a measure of spread are
16
CHAPTER 2. GETTING TO KNOW YOUR DATA
• σ measures spread about the mean and should be considered only when
the mean is chosen as the measure of center.
• σ = 0 only when there is no spread, that is, when all observations have the
same value. Otherwise σ > 0.
Importantly, an observation is unlikely more than several standard deviations
away from the mean. Mathematically, using Chebyshev’s inequality, it can be
shown that at least (1 − k12 ) × 100% of the observations are no more than k
standard deviations from the mean. Therefore, the standard deviation is a good
indicator of the spread of a data set.
The computation of the variance and standard deviation is scalable in large
databases.
2.2.3
Graphic Displays of Basic Statistical Descriptions of
Data
In this section, we study graphic displays of basic statistical descriptions. These
include quantile plots, quantile-quantile plots, histograms, and scatter plots. Such
graphs are helpful for the visual inspection of data, which is useful for data
preprocessing. The first three of these show univariate distributions (i.e., data for
one attribute), while scatter plots show bivariate distributions (that is, involving
two attributes).
Quantile Plot
In this and the following subsections, we cover common graphic displays of data
distributions. A quantile plot is a simple and effective way to have a first
look at a univariate data distribution. First, it displays all of the data for the
given attribute (allowing the user to assess both the overall behavior and unusual
occurrences). Second, it plots quantile information. Quantiles were described
in Section 2.2.2. Let xi , for i = 1 to N , be the data sorted in increasing order
so that x1 is the smallest observation and xN is the largest for some ordinal
or numeric attribute X. Each observation, xi , is paired with a percentage, fi ,
which indicates that approximately fi × 100% of the data are below the value,
xi . We say “approximately” because there may not be a value with exactly a
fraction, fi , of the data below xi . Note that the 0.25 percentage corresponds to
quartile Q1 , the 0.50 percentage is the median, and the 0.75 percentage is Q3 .
Let
i − 0.5
fi =
.
(2.7)
N
1
These numbers increase in equal steps of 1/N , ranging from 2N
(which is slightly
1
above zero) to 1 − 2N (which is slightly below one). On a quantile plot, xi
is graphed against fi . This allows us to compare different distributions based
on their quantiles. For example, given the quantile plots of sales data for two
different time periods, we can compare their Q1 , median, Q3 , and other fi values
at a glance.
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
17
Table 2.1: A set of unit price data for items sold at a branch of
AllElectronics.
Unit price ($)
Count of items sold
275
300
250
..
360
515
540
..
320
270
350
Unit price ($)
40
43
47
..
74
75
78
..
115
117
120
140
120
100
80
60
40
20
0
0.000
0.250
0.500
f-value
0.750
1.000
Figure 2.4: A quantile plot for the unit price data of Table 2.1. NOTE: This
graph must be changed so that Q1, Median, and Q3 denote every 9th point,
respectively. Every 9th point is a quantile. Also, add labels Q1, Median, Q3 as
darkened points in the graph.
Example 2.13 Quantile plot. Figure 2.4 shows a quantile plot for the unit price data of
Table 2.1.
Quantile-Quantile (Q-Q) Plot
A quantile-quantile plot, or q-q plot, graphs the quantiles of one univariate distribution against the corresponding quantiles of another. It is a powerful
visualization tool in that it allows the user to view whether there is a shift in going
from one distribution to another.
Suppose that we have two sets of observations for the attribute or variable
unit price, taken from two different branch locations. Let x1 , . . . , xN be the data
from the first branch, and y1 , . . . , yM be the data from the second, where each
18
CHAPTER 2. GETTING TO KNOW YOUR DATA
120
Branch 2 (unit price $)
110
100
90
80
70
60
50
40
40
50
60
70
80
90
Branch 1 (unit price $)
100
110
120
Figure 2.5: A quantile-quantile plot for unit price data from two different
branches. NOTE: This graph must be changed so that Q1, Median, and Q3
denote every 9th point, respectively. Every 9th point is a quantile. Also, add
labels for Q1, Median, and Q3.
data set is sorted in increasing order. If M = N (i.e., the number of points
in each set is the same), then we simply plot yi against xi , where yi and xi
are both (i − 0.5)/N quantiles of their respective data sets. If M < N (i.e.,
the second branch has fewer observations than the first), there can be only M
points on the q-q plot. Here, yi is the (i − 0.5)/M quantile of the y data, which
is plotted against the (i − 0.5)/M quantile of the x data. This computation
typically involves interpolation.
Example 2.14 Quantile-quantile plot. Figure 2.5 shows a quantile-quantile plot for unit
price data of items sold at two different branches of AllElectronics during a
given time period. Each point corresponds to the same quantile for each data
set and shows the unit price of items sold at branch 1 versus branch 2 for that
quantile. (To aid in comparison, a straight line is shown that represents the case
where, for each given quantile, the unit price at each branch is the same. The
darker points correspond to the data for Q1 , the median, and Q3 , respectively.)
We see, for example, that at Q1 , the unit price of items sold at branch 1 was
slightly less than that at branch 2. In other words, 25% of items sold at branch
1 were less than or equal to $60, while 25% of items at branch 2 were less than or
equal to $64. At the 50th percentile (marked by the median, which is also Q2 ),
we see that 50% of items sold at branch 1 were less than $80, while 50% of items
at branch 2 were less than $90. In general, we note that there is a shift in the
distribution of branch 1 with respect to branch 2 in that the unit prices of items
sold at branch 1 tend to be lower than those at branch 2.
2.2. BASIC STATISTICAL DESCRIPTIONS OF DATA
19
6000
Count of items sold
5000
4000
3000
2000
1000
0
40–59
60–79
80–99
100–119
120–139
Unit Price ($)
Figure 2.6: A histogram for the data set of Table 2.1.
Histograms
Histograms (or frequency histograms) are at least a century old and are
widely used. “Histog” means pole or mast, and “gram” means chart, so a
histogram is a chart of poles. Plotting histograms is a graphical method for
summarizing the distribution of a given attribute, X. If X is nominal, such
as automobile model or item type, then a pole or vertical bar is drawn for each
known value of X. The height of the bar indicates the frequency (i.e., count) of
that X value. The resulting graph is more commonly known as a bar chart.
If X is numeric, the term histogram is preferred. The range of values for
X is partitioned into disjoint consecutive subranges. The subranges, referred
to as buckets, are disjoint subsets of the data distribution for X. The range of
a bucket is known as the width. Typically, the buckets are equi-width. For
example, a price attribute with a value range of $1 to $200 (rounded up to the
nearest dollar) can be partitioned into subranges 1 to 20, 21 to 40, 41 to 60, and
so on. For each subrange, a bar is drawn whose height represents the total count
of items observed within the subrange. Histograms and partitioning rules are
further discussed in Chapter 3 on data reduction.
Example 2.15 Histogram. Figure 2.6 shows a histogram for the data set of Table 2.1, where
buckets are defined by equal-width ranges representing $20 increments and the
frequency is the count of items sold.
Although histograms are widely used, they may not be as effective as the
quantile plot, q-q plot, and boxplot methods in comparing groups of univariate
observations.
20
CHAPTER 2. GETTING TO KNOW YOUR DATA
700
Items sold
600
500
400
300
200
100
0
0
20
40
60
80
100
120
140
Unit price ($)
Figure 2.7: A scatter plot for the data set of Table 2.1.
Figure 2.8: Scatter plots can be used to find (a) positive or (b) negative correlations between attributes.
Scatter Plots and Data Correlation
A scatter plot is one of the most effective graphical methods for determining
if there appears to be a relationship, pattern, or trend between two numeric
attributes. To construct a scatter plot, each pair of values is treated as a pair of
coordinates in an algebraic sense and plotted as points in the plane. Figure 2.7
shows a scatter plot for the set of data in Table 2.1.
The scatter plot is a useful method for providing a first look at bivariate data
to see clusters of points and outliers, or to explore the possibility of correlation
relationships. Two attributes, X, and Y , are correlated if one attribute implies the other. Correlations can be positive, negative, or null (uncorrelated).
Figure 2.8 shows examples of positive and negative correlations between two attributes. If the pattern of plotted points slopes from lower left to upper right,
this means that the values of X increase as the values of Y increase, which suggests a positive correlation (Figure 2.8a)). If the pattern of plotted points slopes
from upper left to lower right, then the values of X increase as the values of
Y decrease, suggesting a negative correlation (Figure 2.8b)). A line of best fit
can be drawn in order to study the correlation between the variables. Statistical
2.3. DATA VISUALIZATION
21
Figure 2.9: Three cases where there is no observed correlation between the two
plotted attributes in each of the data sets.
tests for correlation are given in Chapter 3 on data integration (Equation (3.3)).
Figure 2.9 shows three cases for which there is no correlation relationship between the two attributes in each of the given data sets. Section 2.3.2 shows how
scatter plots can be extended to n attributes, resulting in a scatter plot matrix.
In conclusion, basic data descriptions (such as measures of central tendency
and measures of dispersion), and graphic statistical displays (such as quantile
plots, histograms, and scatter plots) provide valuable insight into the overall behavior of your data. By helping to identify noise and outliers, they are especially
useful for data cleaning.
2.3
Data Visualization
How can we convey data to users effectively? Data visualization aims to communicate data clearly and effectively through graphical representation. Data
visualization has been used extensively in many applications. For example, we
can use data visualization at work for reporting, managing business operations,
and tracking progress of tasks. More popularly, we may take advantage of visualization techniques to discover data relationships that are otherwise not easily
observable by looking at the raw data. Nowadays, people also use data visualization to create fun and interesting graphics.
In this section, we briefly introduce the basic concepts of data visualization.
We start with multidimensional data such as those stored in relational databases.
We discuss several representative approaches, including pixel-oriented techniques,
geometric projection techniques, icon-based techniques, and hierarchical and
graph-based techniques.
We then discuss the visualization of complex data
and relations.
2.3.1
Pixel-Oriented Visualization Techniques
A simple way to visualize the value of a dimension is to use a pixel where the color
of the pixel reflects the dimension’s value. For a data set of m dimensions, pixeloriented techniques create m windows on the screen, one for each dimension.
The m dimension values of a record are mapped to m pixels at the corresponding
22
CHAPTER 2. GETTING TO KNOW YOUR DATA
(a) income
(b) credit limit (c) transaction volume
(d) age
Figure 2.10: Pixel-oriented visualization of four attributes by sorting all customers in income ascending order.
positions in the windows. The colors of the pixels reflect the corresponding
values.
Inside a window, the data values are arranged in some global order shared by
all windows. The global order may be obtained by sorting all data records in a
way that’s meaningful for the task at hand.
Example 2.16 Pixel-oriented visualization. AllElectronics maintains a customer information table, which consists of four dimensions: income, credit limit, transaction
volume, and age. Can we analyze the correlation between income and the other
attributes by visualization?
We can sort all customers in income ascending order, and use this order to lay
out the customer data in the four visualization windows, as shown in Figure 2.10.
The pixel colors are chosen so that the smaller the value is, the lighter the color
is. Using pixel-based visualization, we can easily observe the following: credit
limit increases as income increases; customers whose income is in the middle
range are more likely to purchase more from AllElectronics; there is no clear
correlation between income and age.
In pixel-oriented techniques, data records can also be ordered in a querydependent way. For example, given a point query, we can sort all records in
descending order of similarity to the point query.
Filling a window by laying out the data records in a linear way may not work
well for a wide window. The first pixel in a row is far away from the last pixel
in the previous row, though they are next to each other in the global order.
Moreover, a pixel is next to the one above it in the window, even though the
two are not next to each other in the global order.
To solve this problem,
we can lay out the data records in a space-filling curve to fill the windows. A
space-filling curve is a curve whose range covers the entire n-dimensional unit
hypercube. Since the visualization windows are 2-dimensional, we can use any
2-dimensional space-filling curve. Figure 2.11 shows some frequently used 2dimensional space-filling-curves.
2.3. DATA VISUALIZATION
(a) Hilbert curve
23
(b) Gray code
(c) Z-curve
Figure 2.11: Some frequently used 2-dimensional space-filling curves.
one data record
Dim 6
Dim 6
Dim 1
Dim 5
Dim 5
Dim 1
Dim 2
Dim 2
Dim 4
Dim 4
Dim 3
(a)
Dim 3
(b)
Figure 2.12: The circle segment technique. (a) Representing a data record in
circle segments. (b) Laying out pixels in circle segments.
Note that the windows do not have to be rectangular. For example, the circle
segment technique uses windows in the shape of segments of a circle, as illustrated
in Figure 2.12. This technique can ease the comparison of dimensions because
the dimension windows are located side by side and form a circle.
2.3.2
Geometric Projection Visualization Techniques
A drawback of pixel-oriented visualization techniques is that they cannot help
us much in understanding the distribution of data in a multidimensional space.
For example, they do not show whether there is a dense area in a multidimensional subspace. Geometric projection techniques help users find interesting
projections of multidimensional data sets. The central challenge the geometric
projection techniques try to address is how to visualize a high dimensional space
on a 2-dimensional display.
A scatter plot displays 2-dimensional data points using Cartesian coordinates. A third dimension can be added using different colors or shapes to represent different data points. Figure 2.13 shows an example, where X and Y are
24
CHAPTER 2. GETTING TO KNOW YOUR DATA
Co−location Patterns − Sample Data
80
70
60
Y
50
40
30
20
10
0
0
10
20
30
40
X
50
60
70
80
Figure 2.13: Visualization of a 2-dimensional data set using a scatter plot.
Source: http://www.cs.sfu.ca/ jpei/publications/rareevent-geoinformatica06.pdf.
two spatial attributes and the third dimension is represented by different shapes.
Through this visualization, we can see that points of type “+” and “×” tend to
be co-located.
A 3-dimensional scatter plot uses three axes in a Cartesian coordinate system.
If it also uses color, it can display up to 4-dimensional data points (Figure 2.14).
For data sets whose dimensionality is over four, scatter plots are usually
ineffective. The scatter plot matrix technique is a useful extension to the
scatter plot. For an n-dimensional data set, a scatter plot matrix is an n × n grid
of 2-dimensional scatter plots that provides a visualization of each dimension
with every other dimension. Figure 2.15 shows an example, which visualizes the
Iris data set. The data set consists of 450 samples from each of three species of
Iris flowers. There are five dimensions in the data set: the length and the width
of sepal and petal, and the species.
The scatter-plot matrix becomes less effective as the dimensionality increases.
Another popular technique, called parallel coordinates, can handle higher dimensionality. To visualize n-dimensional data points, the parallel coordinates
technique draws n equally spaced axes, one for each dimension, parallel to one
of the display axes. A data record is represented by a polygonal line that intersects each axis at the point corresponding to the associated dimension value
(Figure 2.16).
A major limitation of the parallel coordinates technique is that it cannot
2.3. DATA VISUALIZATION
25
Figure 2.14: Visualization of a 3-dimensional data set using a scatter plot.
Source: http://upload.wikimedia.org/wikipedia/commons/c/c4/Scatter plot.jpg.
effectively show a data set of many records. Even for a data set of several
thousand records, visual clutter and overlap often reduce the readability of the
visualization and make the patterns hard to find.
2.3.3
Icon-Based Visualization Techniques
Icon-based visualization techniques use small icons to represent multidimensional data values. We look at two popular icon-based techniques – Chernoff
faces and stick figures.
Chernoff faces were introduced in 1973 by statistician Herman Chernoff.
They display multidimensional data of up to eighteen variables (or dimensions)
as a cartoon human face (Figure 2.17). Chernoff faces help reveal trends in the
data. Components of the face, such as the eyes, ears, mouth, and nose, represent
values of the dimensions by their shape, size, placement, and orientation. For
example, dimensions can be mapped to the following facial characteristics: eye
size, eye spacing, nose length, nose width, mouth curvature, mouth width, mouth
openness, pupil size, eyebrow slow, eye eccentricity, and head eccentricity.
Chernoff faces make use of the ability of the human mind to recognize small
differences in facial characteristics and to assimilate many facial characteristics
at once. Viewing large tables of data can be tedious. By condensing the data,
Chernoff faces make the data easier for users to digest. In this way, they facilitate
the visualization of regularities and irregularities present in the data although
their power in relating multiple relationships is limited. Another limitation is
that specific data values are not shown. Furthermore, the features of the face
vary in perceived importance. This means that the similarity of two faces (representing two multidimensional data points) can vary depending on the order in
which dimensions are assigned to facial characteristics. Therefore, this mapping
26
CHAPTER 2. GETTING TO KNOW YOUR DATA
Figure 2.15: Visualization of the Iris data set using scatter plot matrix. Source:
http://support.sas.com/documentation/cdl/en/grstatproc/61948/HTML/default
/images/gsgscmat.gif.
should be carefully chosen. Eye size and eyebrow slant have been found to be
important.
Asymmetrical Chernoff faces were proposed as an extension to the original
technique. Since a face has vertical symmetry (along the y-axis), the left and
right side of a face are identical, which wastes space. Asymmetrical Chernoff faces
double the number of facial characteristics, thus allowing up to 36 dimensions to
be displayed.
The stick figure visualization technique maps multidimensional data to 5piece stick figures, where each figure has four limbs and a body. Two dimensions
are mapped to the display (x and y) axes and the remaining dimensions are
mapped to the angle and/or length of the limbs. Figure 2.18 shows census data,
where age and income are mapped to the display axes, and the remaining dimensions (gender, education, and so on) are mapped to stick figures. If the
data items are relatively dense with respect to the two display dimensions, the
resulting visualization shows texture patterns, reflecting data trends.
2.3.4
Hierarchical Visualization Techniques
The visualization techniques discussed so far focus on visualizing multiple dimensions simultaneously. However, for a large data set of high dimensionality,
it would be difficult to visualize all dimensions at the same time.
Hierar-
2.3. DATA VISUALIZATION
Figure 2.16:
Visualization using parallel coordinates.
http://www.stat.columbia.edu/ cook/movabletype/mlm/mdg1.png.
27
Source:
Figure 2.17: Chernoff faces. Each face represents an n-dimensional data point
(n ≤ 18).
chical visualization techniques partition all dimensions into subsets (that is,
subspaces). The subspaces are visualized in a hierarchical manner.
“Worlds-within-Worlds”, also known as n-Vision, is a representative hierarchical visualization method. Suppose we want to visualize a 6-dimensional
data set, where the dimensions are F, X1 , . . . , X5 . We want to observe how dimension F changes with respect to the other dimensions. We can first fix the
values of dimensions X3 , X4 , X5 to some selected values, say, c3 , c4 , c5 . We can
then visualize F, X1 , X2 using a 3-dimensional plot, called a world, as shown
in Figure 2.19. The position of the origin of the inner world is located at the
point (c3 , c4 , c5 ) in the outer world, which is another 3-dimensional plot using
dimensions X3 , X4 , X5 . A user can interactively change, in the outer world,
the location of the origin of the inner world. The user then views the resulting
changes of the inner world. Moreover, a user can vary the dimensions used in the
inner world and the outer world. Given more dimensions, more levels of worlds
28
CHAPTER 2. GETTING TO KNOW YOUR DATA
Figure 2.18: Census data represented using stick figures. Source: G. Grinstein,
University of Massachusetts at Lovell.
can be used, which is why the method is called “worlds-within-worlds”.
As another example of hierarchical visualization methods, treemaps display
hierarchical data as a set of nested rectangles. For example, Figure 2.20 shows
a treemap visualizing Google news stories. All news stories are organized into
seven categories, each shown in a large rectangle of a unique color. Within each
category (that is, each rectangle at the top level), the news stories are further
partitioned into smaller subcategories.
2.3.5
Visualizing Complex Data and Relations
In early days, visualization techniques were mainly for numeric data. Recently,
more and more non-numeric data, such as text and social networks, have become
available. Visualizing and analyzing such data attracts a lot of interest.
There are many new visualization techniques dedicated to these kinds of data.
For example, many people on the Web tag various objects such as pictures, blog
entries, and product reviews. A tag cloud is a visualization of statistics of
user-generated tags. Often, in a tag cloud, tags are listed alphabetically or in
a user-preferred order. The importance of a tag is indicated by the font size or
color. Figure 2.21 shows a tag cloud for visualizing the popular tags used in a
Web site.
Tag clouds are often used in two ways. First, in a tag cloud for a single
item, we can use the size of a tag to represent the number of times that the tag is
applied to this item by different users. Second, when visualizing the tag statistics
on multiple items, we can use the size of a tag to represent the number of items
that the tag has been applied to, that is, the popularity of the tag.
In addition to complex data, complex relations among data entries also raise
challenges for visualization. For example, Figure 2.22 uses a disease influence
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
29
Figure 2.19: “Worlds-within-World” (also known as n-Vision).
Source:
http://graphics.cs.columbia.edu/projects/AutoVisual/images/1.dipstick.5.gif.
graph to visualize the correlations between diseases. The nodes in the graph
are diseases, and the size of each node is proportional to the prevalence of the
corresponding disease. Two nodes are linked by an edge if the corresponding
diseases have a strong correlation. The width of an edge is proportional to the
strength of the correlation pattern of the two corresponding diseases.
In summary, visualization provides effective tools to explore data. We have
introduced some popular methods and the essential ideas behind them. There
are many existing tools and methods. Moreover, visualization can be used in
data mining in various aspects. In addition to visualizing data, visualization
can be used to represent the data mining process, the patterns obtained from a
mining method, and user interaction with the data. Visual data mining is an
important research and development direction.
2.4
Measuring Data Similarity and Dissimilarity
In data mining applications such as clustering, outlier analysis, and nearestneighbor classification, we need ways of assessing how alike objects are, or how
unlike they are in comparison to one another. For example, a store may like to
search for clusters of customer objects, resulting in groups of customers with similar characteristics (e.g., similar income, area of residence, and age). Such information can then be used for marketing. A cluster is a collection of data objects
such that the objects within a cluster are similar to one another and dissimilar
to the objects in other clusters. Outlier analysis also employs clustering-based
techniques to identify potential outliers as objects that are highly dissimilar to
30
CHAPTER 2. GETTING TO KNOW YOUR DATA
Figure
2.20:
Newsmap:
Use
of
treemaps
to
visualize
Google
news
headline
stories.
Source:
http://www.cs.umd.edu/class/spring2005/cmsc838s/viz4all/ss/newsmap.png.
others. Knowledge of object similarities can also be used in nearest-neighbor
classification schemes where a given object (such as a patient ) is assigned a class
label (relating to, say, a diagnosis) based on its similarity towards other objects
in the model.
This section presents measures of similarity and dissimilarity. Similarity and
dissimilarity measures are referred to as measures of proximity. Similarity and
dissimilarity are related. A similarity measure for two objects, i and j, will typically return the value 0 if the objects are unalike. The higher the similarity
value, the greater the similarity between objects. (Typically, a value of 1 indicates complete similarity, that is, that the objects are identical.) A dissimilarity
measure works the opposite way. It returns a value of 0 if the objects are the
same (and therefore, far from being dissimilar). The higher the dissimilarity
value, the more dissimilar the two objects are.
We begin in Section 2.4.1 by presenting two data structures that are commonly used in the above types of applications. These are the data matrix (used
to store the data objects) and the dissimilarity matrix (used to store dissimilarity
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
31
Figure 2.21: Using a tag cloud to visualize the popular tags used in a Web
site. Source: a snapshot of http://www.flickr.com/photos/tags/ as of January
23, 2010.
values for pairs of objects). We also switch to a different notation for data objects than previously used in this chapter since now we are dealing with objects
described by more than one attribute. We then discuss how object dissimilarity
can be computed for objects described by nominal attributes (Section 2.4.2);
by binary attributes (Section 2.4.3); by numeric attributes (Section 2.4.4); by
ordinal attributes (Section 2.4.5); or by combinations of these attribute types
(Section 2.4.6). Section 2.4.7 provides similarity measures for very long and
sparse data vectors, such as term-frequency vectors representing documents in
information retrieval. Knowing how to compute dissimilarity is useful in studying attributes and will also be referenced in later topics on clustering (Chapters
10 and 11), outlier analysis (Chapter 12), and nearest-neighbor classification
(Chapter 8).
2.4.1
Data Matrix vs. Dissimilarity Matrix
In Section 2.2, we looked at ways of studying the central tendency, dispersion, and spread of observed values for some attribute X. Our objects there
were 1-dimensional, that is, described by a single attribute. In this section,
we talk about objects described by multiple attributes. Therefore, we need a
change in notation. Suppose that we have n objects (such as persons, items,
or courses) described by p attributes (also called measurements or features),
such as age, height, weight, or gender. The objects are x1 = (x11 , x12 , . . . , x1p ),
x2 = (x21 , x22 , . . . , x2p ), and so on, where xij is the value for object xi of the j th
32
CHAPTER 2. GETTING TO KNOW YOUR DATA
Figure 2.22: The disease influence graph of people of age at least 20 in the
NHANES data set.
attribute. For brevity, we hereafter refer to object xi as object i. The objects
may be tuples in a relational database, and are also referred to as data samples
or feature vectors.
Main memory-based clustering and nearest-neighbor algorithms typically operate on either of the following two data structures.
• Data matrix (or object-by-attribute structure): This structure stores the
n data objects in the form of a relational table, or n-by-p matrix (n objects
×p attributes):


x11 · · · x1f · · · x1p
 ··· ··· ··· ··· ··· 


 xi1 · · · xif · · · xip 
(2.8)


 ··· ··· ··· ··· ··· 
xn1 · · · xnf · · · xnp
Each row corresponds to an object. As part of our notation, we may use f
to index through the p attributes.
• Dissimilarity matrix (or object-by-object structure): This stores a collection of proximities that are available for all pairs of n objects. It is often
represented by an n-by-n table:


0

 d(2, 1)
0



 d(3, 1) d(3, 2) 0
(2.9)




..
..
..


.
.
.
d(n, 1) d(n, 2) · · · · · · 0
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
33
where d(i, j) is the measured dissimilarity or “difference” between objects
i and j. In general, d(i, j) is a nonnegative number that is close to 0 when
objects i and j are highly similar or “near” each other, and becomes larger
the more they differ. Note that d(i, i) = 0, that is, the difference between
an object and itself is 0. Furthermore, d(i, j) = d(j, i). (For readability,
we do not show the d(j, i) entries; the matrix is symmetric.) Measures of
dissimilarity are discussed throughout the remainder of this chapter.
Measures of similarity can often be expressed as a function of measures of
dissimilarity. For example, for nominal data,
sim(i, j) = 1 − d(i, j)
(2.10)
where sim(i, j) is the similarity between objects i and j. Throughout the rest
of this chapter, we will also comment on measures of similarity.
A data matrix is made up of two entities or “things”, namely rows (for objects) and columns (for attributes). Therefore, the data matrix is often called
a two-mode matrix. The dissimilarity matrix contains one kind of entity (dissimilarities) and so is called a one-mode matrix. Many clustering and nearestneighbor algorithms operate on a dissimilarity matrix. Data in the form of a
data matrix can be transformed into a dissimilarity matrix before applying such
algorithms.
2.4.2
Proximity Measures for Nominal Attributes
A nominal attribute can take on two or more states (Section 2.1.2). For example,
map color is a nominal attribute that may have, say, five states: red, yellow,
green, pink, and blue.
Let the number of states of a nominal attribute be M . The states can be
denoted by letters, symbols, or a set of integers, such as 1, 2, . . . , M . Notice that
such integers are used just for data handling and do not represent any specific
ordering.
“How is dissimilarity computed between objects described by nominal attributes?” The dissimilarity between two objects i and j can be computed
based on the ratio of mismatches:
d(i, j) =
p−m
,
p
(2.11)
where m is the number of matches (i.e., the number of attributes for which i and
j are in the same state), and p is the total number of attributes describing the
objects. Weights can be assigned to increase the effect of m or to assign greater
weight to the matches in attributes having a larger number of states.
Example 2.17 Dissimilarity between nominal attributes. Suppose that we have the sample data of Table 2.2, except that only the object-identifier and the attribute
34
CHAPTER 2. GETTING TO KNOW YOUR DATA
test-1 are available, where test-1 is nominal. (We will use test-2 and test-3 in
later examples.) Let’s compute the dissimilarity matrix (2.9), that is,


0
 d(2, 1)

0


 d(3, 1) d(3, 2)

0
d(4, 1) d(4, 2) d(4, 3) 0
Since here we have one nominal attribute, test-1, we set p = 1 in Equation (2.11)
so that d(i, j) evaluates to 0 if objects i and j match, and 1 if the objects differ.
Thus, we get

0
 1 0

 1 1
0 1




0
1 0
From this, we see that all objects are dissimilar to one another except objects
1 and 4 (that is, d(4, 1) = 0)).
Alternatively, similarity can be computed as
sim(i, j)
= 1 − d(i, j) =
m
.
p
(2.12)
Proximity between objects described by nominal attributes can be computed
using an alternative encoding scheme. Nominal attributes can be encoded by
asymmetric binary attributes by creating a new binary attribute for each of the M
states. For an object with a given state value, the binary attribute representing
that state is set to 1, while the remaining binary attributes are set to 0. For
example, to encode the nominal attribute map color, a binary attribute can be
created for each of the five colors listed above. For an object having the color
yellow, the yellow attribute is set to 1, while the remaining four attributes are
set to 0. Proximity measures for this form of encoding can be calculated using
the methods discussed in the following section.
Table 2.2: A sample data table containing attributes
of mixed type.
object
identifier
1
2
3
4
test-1
(nominal )
test-2
(ordinal)
code-A
code-B
code-C
code-A
excellent
fair
good
excellent
test-3
(numeric )
45
22
64
28
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
2.4.3
35
Proximity Measures for Binary Attributes
Let’s look at dissimilarity and similarity measures for objects described by either
symmetric or asymmetric binary attributes.
Recall that a binary attribute has only two states: 0 or 1, where 0 means
that the attribute is absent, and 1 means that it is present (Section 2.1.3).
Given the attribute smoker describing a patient, for instance, 1 indicates that
the patient smokes, while 0 indicates that the patient does not. Treating binary
attributes as if they are numeric can be misleading. Therefore, methods specific
to binary data are necessary for computing dissimilarities.
“So, how can we compute the dissimilarity between two binary attributes?”
One approach involves computing a dissimilarity matrix from the given binary
data. If all binary attributes are thought of as having the same weight, we have
the 2-by-2 contingency table of Table 2.3, where q is the number of attributes
that equal 1 for both objects i and j, r is the number of attributes that equal 1
for object i but that are 0 for object j, s is the number of attributes that equal
0 for object i but equal 1 for object j, and t is the number of attributes that
equal 0 for both objects i and j. The total number of attributes is p, where p =
q + r + s + t.
Recall that for symmetric binary attributes, each state is equally valuable.
Dissimilarity that is based on symmetric binary attributes is called symmetric
binary dissimilarity. If objects i and j are described by symmetric binary
attributes, then the dissimilarity between i and j is
d(i, j) =
r+s
.
q+r+s+t
(2.13)
For asymmetric binary attributes, the two states are not equally important,
such as the positive (1) and negative (0) outcomes of a disease test. Given two
asymmetric binary attributes, the agreement of two 1s (a positive match) is then
considered more significant than that of two 0s (a negative match). Therefore,
such binary attributes are often considered “monary” (as if having one state).
The dissimilarity based on such attributes is called asymmetric binary dissimilarity, where the number of negative matches, t, is considered unimportant
and thus is ignored in the computation, as shown in Equation (2.14).
d(i, j) =
r+s
.
q+r+s
(2.14)
Table 2.3: A contingency table for binary attributes.
object i
1
0
sum
object j
1
q
s
q+s
0
r
t
r+t
sum
q+r
s+t
p
36
CHAPTER 2. GETTING TO KNOW YOUR DATA
Complementarily, we can measure the difference between two binary attributes based on the notion of similarity instead of dissimilarity. For example, the asymmetric binary similarity between the objects i and j can be
computed as,
sim(i, j) =
q
= 1 − d(i, j).
q+r+s
(2.15)
The coefficient sim(i, j) of Equation (2.15) is called the Jaccard coefficient
and is popularly referenced in the literature.
When both symmetric and asymmetric binary attributes occur in the same
data set, the mixed attributes approach described in Section 2.4.6 can be applied.
Example 2.18 Dissimilarity between binary attributes. Suppose that a patient record
table (Table 2.4) contains the attributes name, gender, fever, cough, test-1, test2, test-3, and test-4, where name is an object identifier, gender is a symmetric
attribute, and the remaining attributes are asymmetric binary.
For asymmetric attribute values, let the values Y (yes) and P (positive) be
set to 1, and the value N (no or negative) be set to 0. Suppose that the distance
between objects (patients) is computed based only on the asymmetric attributes.
According to Equation (2.14), the distance between each pair of the three patients,
Jack, Mary, and Jim, is
d(Jack, Jim)
=
1+1
1+1+1
= 0.67
d(Jack, Mary)
=
0+1
2+0+1
= 0.33
d(Jim, Mary)
=
1+2
1+1+2
= 0.75
These measurements suggest that Jim and Mary are unlikely to have a similar
disease because they have the highest dissimilarity value among the three pairs.
Of the three patients, Jack and Mary are the most likely to have a similar disease.
Table 2.4: A relational table where patients are described by binary
attributes.
name
Jack
Jim
Mary
..
.
gender
M
M
F
..
.
fever
Y
Y
Y
..
.
cough
N
Y
N
..
.
test-1
P
N
P
..
.
test-2
N
N
N
..
.
test-3
N
N
P
..
.
test-4
N
N
N
..
.
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
2.4.4
37
Dissimilarity on Numeric Data: Minkowski Distance
In this section, we describe distance measures that are commonly used for computing the dissimilarity of objects described by numeric attributes. These measures include the Euclidean, Manhattan and Minkowski distances.
In some cases, the data are normalized before applying distance calculations.
This involves transforming the data to fall within a smaller or common range,
such as [−1, 1] or [0.0,1.0]. Consider a height attribute, for example, which could
be measured in either meters or inches. In general, expressing an attribute in
smaller units will lead to a larger range for that attribute, and thus tend to give
such attributes greater effect or “weight”. Normalizing the data attempts to
give all attributes an equal weight. It may or may not be useful in a particular
application. Methods for normalizing data are discussed in detail in Chapter 3
on data preprocessing.
The most popular distance measure is Euclidean distance (i.e., straight
line or “as the crow flies” distance).
Let i = (xi1 , xi2 , . . . , xip ) and j =
(xj1 , xj2 , . . . , xjp ) be two objects described by p numeric attributes. The Euclidean distance between objects i and j is defined as
q
d(i, j) = (xi1 − xj1 )2 + (xi2 − xj2 )2 + · · · + (xip − xjp )2 .
(2.16)
Another well-known measure is Manhattan (or city block) distance,
named so because it is the distance in blocks between any two points in a city
(such as 2 blocks down and 3 blocks over for a total of 5 blocks). It is defined as
d(i, j) = |xi1 − xj1 | + |xi2 − xj2 | + · · · + |xip − xjp |.
(2.17)
Both the Euclidean distance and Manhattan distance satisfy the following mathematical properties:
Non-negativity: d(i, j) ≥ 0: Distance is a nonnegative number.
Identity of indiscernibles: d(i, i) = 0: The distance of an object to itself is 0.
Symmetry: d(i, j) = d(j, i): Distance is a symmetric function.
Triangle inequality: d(i, j) ≤ d(i, k) + d(k, j): Going directly from object i
to object j in space is no more than making a detour over any other object
k (triangular inequality).
A measure that satisfies these conditions is known as metric. Please note that
the non-negativity property is implied by the other three properties.
Example 2.19 Euclidean distance and Manhattan distance. Let x1 = (1, 2) and x2 = (3,
5) represent
two objects as in Figure 2.23. The Euclidean distance between the
√
two is 22 + 32 = 3.61. The Manhattan distance between the two is 2 + 3 = 5.
38
CHAPTER 2. GETTING TO KNOW YOUR DATA
Minkowski distance is a generalization of both Euclidean distance and Manhattan distance. It is defined as
q
d(i, j) = h |xi1 − xj1 |h + |xi2 − xj2 |h + · · · + |xip − xjp |h ,
(2.18)
The original before taking the student comment. where h is a real number such
that h ≥ 1. (Such a distance is also called Lp norm in some literature, where the
symbol p refers to our notation of h. We have kept p as the number of attributes
to be consistent with the rest of this chapter.) It represents the Manhattan
distance when h = 1 (i.e., L1 norm) and Euclidean distance when h = 2 (i.e., L2
norm).
The supremum distance (also referred to as Lmax , L∞ norm, and the
Chebyshev distance) is a generalization of Minkowski distance for h → ∞. To
compute it, we find the attribute f that gives the maximum difference in values
between the two objects. This difference is the supremum distance, defined more
formally as:

 h1
p
X
p
(2.19)
|xif − xjf |h  = max |xif − xjf |.
d(i, j) = lim 
h→∞
f
f =1
The L∞ norm is also known as the uniform norm.
Example 2.20 Supremum distance. Let’s use the same two objects, x1 = (1, 2) and x2 = (3,
5), as in Figure 2.23. The second attribute gives the greatest difference is values
for the objects, which is 5 − 2 = 3. This is the supremum distance between both
objects.
If each attribute is assigned a weight according to its perceived importance,
the weighted Euclidean distance can be computed as
q
d(i, j) = w1 |xi1 − xj1 |2 + w2 |xi2 − xj2 |2 + · · · + wm |xip − xjp |2 .
(2.20)
x2 = (3,5)
5
4
3
Euclidean distance
= (22 + 32)1/2 = 3.61
3
2
x1 = (1,2)
Manhattan distance
=2+3=5
2
1
1
2
3
Figure 2.23: Euclidean and Manhattan distances between two objects. NOTE:
Add supremum distance = 5 − 2 = 3.
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
39
Weighting can also be applied to other distance measures as well.
2.4.5
Proximity Measures for Ordinal Attributes
The values of an ordinal attribute have a meaningful order or ranking about
them, yet the magnitude between successive values is unknown (Section 2.1.4).
An example includes the sequence small, medium, large for a size attribute.
Ordinal attributes may also be obtained from the discretization of numeric
attributes by splitting the value range into a finite number of categories. These
categories are organized into ranks. That is, the range of a numeric attribute
can be mapped to an ordinal attribute f having Mf states. For example, the
range of interval-scaled attribute temperature (in Celsius) can be organized
into the following states: −30 to −10, −10 to 10, 10 to 30, representing
the categories cold temperature, moderate temperature, and warm temperature,
respectively. Let the number of possible states that an ordinal attribute can
have be M . These ordered states define the ranking 1, . . . , Mf .
“How are ordinal attributes handled?” The treatment of ordinal attributes
is quite similar to that of numeric attributes when computing the dissimilarity
between objects. Suppose that f is an attribute from a set of ordinal attributes
describing n objects. The dissimilarity computation with respect to f involves
the following steps:
1. The value of f for the ith object is xif , and f has Mf ordered states, representing the ranking 1, . . . , Mf . Replace each xif by its corresponding
rank, rif ∈ {1, . . . , Mf }.
2. Since each ordinal attribute can have a different number of states, it is
often necessary to map the range of each attribute onto [0.0,1.0] so that
each attribute has equal weight. We perform such data normalization by
replacing the rank rif of the ith object in the f th attribute by
zif =
rif − 1
.
Mf − 1
(2.21)
3. Dissimilarity can then be computed using any of the distance measures
described in Section 2.4.4 for numeric attributes, using zif to represent
the f value for the ith object.
Example 2.21 Dissimilarity between ordinal attributes. Suppose that we have the
sample data of Table 2.2, except that this time only the object-identifier and
the continuous ordinal attribute, test -2, are available. There are three states
for test -2, namely fair, good, and excellent, that is Mf = 3. For step 1, if
we replace each value for test -2 by its rank, the four objects are assigned the
ranks 3, 1, 2, and 3, respectively. Step 2 normalizes the ranking by mapping
rank 1 to 0.0, rank 2 to 0.5, and rank 3 to 1.0. For step 3, we can use,
40
CHAPTER 2. GETTING TO KNOW YOUR DATA
say, the Euclidean distance (Equation (2.16)), which results in the following
dissimilarity matrix:


0
 1.0 0



 0.5 0.5 0

0 1.0 0.5 0
Therefore, objects 1 and 2 are the most dissimilar, as are objects 2 and 4 (that
is, d(2, 1) = 1.0 and d(4, 2) = 1.0). This makes intuitive sense since objects 1
and 4 are both excellent. Object 2 is fair, which is at the opposite end of the
range of values for test-2.
Similarity values for ordinal attributes can be interpreted from dissimilarity
as sim(i, j) = 1 − d(i, j).
2.4.6
Dissimilarity for Attributes of Mixed Types
Sections 2.4.2 to 2.4.5 discussed how to compute the dissimilarity between
objects described by attributes of the same type, where these types may be
either nominal, symmetric binary, asymmetric binary, numeric, and ordinal.
However, in many real databases, objects are described by a mixture of attribute types. In general, a database can contain all of the attribute types
listed above.
“So, how can we compute the dissimilarity between objects of mixed attribute
types?” One approach is to group each type of attribute together, performing
separate data mining (e.g., clustering) analysis for each type. This is feasible
if these analyses derive compatible results. However, in real applications, it is
unlikely that a separate analysis per attribute type will generate compatible
results.
A more preferable approach is to process all attribute types together, performing a single analysis. One such technique combines the different attributes
into a single dissimilarity matrix, bringing all of the meaningful attributes onto
a common scale of the interval [0.0,1.0].
Suppose that the data set contains p attributes of mixed type. The dissimilarity d(i, j) between objects i and j is defined as
d(i, j) =
(f )
(f ) (f )
f =1 δij dij
,
Pp
(f )
δ
ij
f =1
Pp
(2.22)
where the indicator δij = 0 if either (1) xif or xjf is missing (i.e., there is
no measurement of attribute f for object i or object j), or (2) xif = xjf = 0
(f )
and attribute f is asymmetric binary; otherwise, δij = 1. The contribution
(f )
of attribute f to the dissimilarity between i and j, that is, dij , is computed
dependent on its type:
2.4. MEASURING DATA SIMILARITY AND DISSIMILARITY
(f )
|x
−x
41
|
if
jf
• If f is numeric: dij = maxh xhf
−minh xhf , where h runs over all nonmissing objects for attribute f .
(f )
(f )
• If f is nominal or binary: dij = 0 if xif = xjf ; otherwise dij = 1.
• If f is ordinal: compute the ranks rif and zif =
numeric.
rif −1
Mf −1 ,
and treat zif as
The above steps are identical to what we have already seen for each of the
individual attribute types. The only difference is for numeric attributes, where
we normalize so that the values map to the interval [0.0,1.0]. Thus, the dissimilarity between objects can be computed even when the attributes describing
the objects are of different types.
Example 2.22 Dissimilarity between attributes of mixed type. Let’s compute a dissimilarity matrix for the objects of Table 2.2. Now we will consider all of
the attributes, which are of different types. In Examples 2.4.2 and 2.4.5, we
worked out the dissimilarity matrices for each of the individual attributes. The
procedures we followed for test -1 (which is nominal) and test -2 (which is ordinal) are the same as outlined above for processing attributes of mixed types.
Therefore, we can use the dissimilarity matrices obtained for test -1 and test -2
later when we compute Equation (2.22). First, however, we need to compute
the dissimilarity matrix for the third attribute, test-3 (which is numeric). That
(3)
is, we must compute dij . Following the case for numeric attributes, we let
maxh xh = 64 and minh xh = 22. The difference between the two is used in
Equation (2.22) to normalize the values of the dissimilarity matrix.
The
resulting dissimilarity matrix for test -3 is:


0
 0.55

0


 0.45 1.00

0
0.40 0.14 0.86 0
We can now use the dissimilarity matrices for the three attributes in our compu(f )
tation of Equation (2.22). The indicator δij = 1 for each of the three attributes,
= 0.65. The resulting disf . We get, for example, d(3, 1) = 1(1)+1(0.50)+1(0.45)
3
similarity matrix obtained for the data described by the three attributes of mixed
types is:


0
 0.85

0


 0.65 0.83

0
0.13 0.71 0.79 0
If we go back and look at Table 2.2, we can intuitively guess that objects 1
and 4 are the most similar, based on their values for test -1 and test -2. This is
42
CHAPTER 2. GETTING TO KNOW YOUR DATA
Document
Document1
Document2
Document3
Document4
Table 2.5: A document vector or term-frequency vector.
teamcoach hockey baseball soccer penalty score win
5
3
0
0
0
0
7
1
3
2
0
0
0
0
2
0
2
1
1
1
0
1
0
2
0
0
0
2
2
1
3
0
confirmed by the dissimilarity matrix, where d(4, 1) is the lowest value for any
pair of different objects. Similarly, the matrix indicates that objects 1 and 2
are the least similar.
2.4.7
Cosine Similarity
A document can be represented by thousands of attributes, each recording the
frequency of a particular word (such as keywords) or phrase in the document.
Thus, each document is an object represented by a what is called a termfrequency vector. For example, in Table 2.5, we see that Document1 contains
five instances of the word team, while hockey occurs three times. The word
coach is absent from the entire document, as indicated by a count value of
0. Such data can be highly asymmetric. Term-frequency vectors are typically
very long and sparse (that is, having many 0 values). Applications using such
structures include information retrieval, text document clustering, biological
taxonomy, and gene feature mapping. The traditional distance measures that
we have studied in this chapter do not work well for such sparse numeric data.
For example, two term-frequency vectors may have many 0 values in common,
meaning that the corresponding documents do not share many words, but this
does not make them similar. We need a measure that will focus on the words
that the two document do have in common, and the occurrence frequency of
such words. In other words, we need a measure for numeric data that ignores
0-matches.
Cosine similarity is a measure of similarity that can be used to compare
documents, or, say, give a ranking of documents with respect to a given vector
of query words. Let x and y be two vectors for comparison. Using the cosine
measure as a similarity function, we have:
sim(x, y) =
x·y
,
||x||||y||
(2.23)
where ||x|| is the Euclidean norm of vector x = (x1 , x2 , . . . , xp ), defined as
q
x21 + x22 + . . . + x2p . Conceptually, it is the length of the vector. Similarly,
||y|| is the Euclidean norm of vector y.
The measure computes the cosine
of the angle between vectors x and y. A cosine value of 0 means that the
two vectors are at 90 degrees to each other (orthogonal) and have no match.
loss
season
0
0
0
3
0
1
0
0
2.5. SUMMARY
43
The closer the cosine value to 1, the smaller the angle and the greater the
match between vectors. Note that because the cosine similarity measure does
not obey all of the properties of Section 2.4.4 defining metric measures, it is
therefore referred to as a nonmetric measure.
Example 2.23 Cosine similarity between two term-frequency vectors. Suppose that
x and y are the first two term-frequency vectors of Table 2.5. That is, x =
(5, 0, 3, 0, 2, 0, 0, 2, 0, 0) and y = (3, 0, 2, 0, 1, 1, 0, 1, 0, 1). How similar are x and
y? Using Equation (2.23) to compute the cosine similarity between the two
vectors, we get:
xt · y
=
||x|| =
||y|| =
sim(x, y) =
5×3+0×0+3×2+0×0+2×1+0×1+0×0+2×1+
0 × 0 + 0 × 1 = 25
p
52 + 02 + 32 + 02 + 22 + 02 + 02 + 22 + 02 + 02 = 6.48
p
32 + 02 + 22 + 02 + 12 + 12 + 02 + 12 + 02 + 12 = 4.12
0.94
Therefore, if we were using the cosine similarity measure to compare these
documents, they would be considered quite similar.
When attributes are binary-valued, the cosine similarity function can be
interpreted in terms of shared features or attributes. Suppose an object x
possesses the ith attribute if xi = 1. Then xt · y is the number of attributes
possessed (that is, shared) by both x and y, and |x||y| is the geometric mean of
the number of attributes possessed by x and the number possessed by y. Thus
sim(x, y) is a measure of relative possession of common attributes.
A simple variation of cosine similarity for the above scenario is
sim(x, y) =
x·y
x·x+y·y−x·y
(2.24)
which is the ratio of the number of attributes shared by x and y to the number
of attributes possessed by x or y. This function, known as the Tanimoto
coefficient or Tanimoto distance, is frequently used in information retrieval
and biology taxonomy.
2.5
Summary
• Data sets are made up of data objects. A data object represents an
entity. Data objects are described by attributes. Attributes can be
nominal, binary, ordinal, or numeric.
• The values of a nominal (or categorical) attribute are symbols or
names of things, where each value represents some kind of category,
code, or state.
44
CHAPTER 2. GETTING TO KNOW YOUR DATA
Binary attributes are nominal attributes with only two possible states
(such as 1 and 0, or true and false). If the two states are equally important, the attribute is a symmetric binary attribute, otherwise it is an
asymmetric binary attribute.
• An ordinal attribute is an attribute whose possible values have a meaningful order or ranking among them, but the magnitude between successive values is not known.
• A numeric attribute is quantitative, that is, it is a measurable quantity, represented in integer or real values. Numeric attribute types can
be interval-scaled or ratio-scaled. The values of an interval-scaled attribute are measured in fixed and equal units. Ratio-scaled attributes
are numeric attributes with an inherent zero-point. Measurements are
ratio-scaled in that we can speak of values as being an order of magnitude larger than the unit of measurement.
• Basic statistical descriptions provide the analytical foundation for
data preprocessing. The basic statistical measures for data summarization include mean, weighted mean, median, and mode for measuring the
central tendency of data; and range, quantiles, quartiles, interquartile
range, variance, and standard deviation for measuring the dispersion of
data. Graphical representations, such as boxplots, quantile plots, quantilequantile plots, histograms, and scatter plots facilitate visual inspection of
the data and are thus useful for data preprocessing and mining.
• Data visualization techniques may be pixel-oriented, geometric-based,
icon-based, or hierarchical. These methods apply to multidimensional
relational data. Additional techniques have been proposed for the visualization of complex data, such as text and social networks.
• Measures of object similarity and dissimilarity are used in data mining applications such as clustering, outlier analysis, and nearest-neighbor
classification. Such measures of proximity can be computed for each of
the attribute types studied in this chapter, or combinations of such attributes. Examples include the Jaccard coefficient for asymmetric binary
attributes; and Euclidean, Manhattan, Minkowski, and supremum distances for numeric attributes. For applications involving sparse numeric
data vectors, such as term-frequency vectors, the cosine measure and the
Tanimoto coefficient are often used in the assessment of similarity.
2.6
Exercises
1. Give three additional commonly used statistical measures (i.e., not illustrated in this chapter) for the characterization of data dispersion, and
discuss how they can be computed efficiently in large databases.
2. Suppose that the data for analysis includes the attribute age. The age
values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20,
20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46,
52, 70.
(a) What is the mean of the data? What is the median?
(b) What is the mode of the data ? Comment on the data’s modality
(i.e., bimodal, trimodal, etc.).
(c) What is the midrange of the data?
(d) Can you find (roughly) the first quartile (Q1) and the third quartile
(Q3) of the data?
(e) Give the five-number summary of the data.
(f) Show a boxplot of the data.
(g) How is a quantile-quantile plot different from a quantile plot ?
3. Suppose that the values for a given set of data are grouped into intervals.
The intervals and corresponding frequencies are as follows.
age
1–5
6–15
16–20
21–50
51–80
81–110
frequency
200
450
300
1500
700
44
Compute an approximate median value for the data.
4. Suppose a hospital tested the age and body fat data for 18 randomly
selected adults with the following result
age
%fat
age
%fat
23
9.5
52
34.6
23
26.5
54
42.5
27
7.8
54
28.8
27
17.8
56
33.4
39
31.4
57
30.2
41
25.9
58
34.1
47
27.4
58
32.9
49
27.2
60
41.2
50
31.2
61
35.7
(a) Calculate the mean, median and standard deviation of age and %fat.
(b) Draw the boxplots for age and %fat.
(c) Draw a scatter plot and a q-q plot based on these two variables.
46
CHAPTER 2. GETTING TO KNOW YOUR DATA
5. Briefly outline how to compute the dissimilarity between objects described by the following:
(a) Nominal attributes
(b) Asymmetric binary attributes
(c) Numeric attributes
(d) Term-frequency vectors
6. Given two objects represented by the tuples (22, 1, 42, 10) and (20, 0,
36, 8):
(a) Compute the Euclidean distance between the two objects.
(b) Compute the Manhattan distance between the two objects.
(c) Compute the Minkowski distance between the two objects, using
q = 3.
(d) Compute the supremum distance between the two objects.
7. The median is one of the most important holistic measures in data analysis. Propose several methods for median approximation. Analyze their
respective complexity under different parameter settings and decide to
what extent the real value can be approximated. Moreover, suggest a
heuristic strategy to balance between accuracy and complexity and then
apply it to all methods you have given.
8. It is important to define or select similarity measures in data analysis.
However, there is no commonly-accepted subjective similarity measure.
Results can vary depending on the similarity measures used. Nonetheless, seemingly different similarity measures may be equivalent after some
transformation.
Suppose we have the following two-dimensional data set:
x1
x2
x3
x4
x5
A1
1.5
2
1.6
1.2
1.5
A2
1.7
1.9
1.8
1.5
1.0
(a) Consider the data as two-dimensional data points. Given a new
data point, x = (1.4, 1.6) as a query, rank the database points based
on similarity with the query using Euclidean distance, Manhattan
distance, supremum distance, and cosine similarity.
(b) Normalize the data set to make the norm of each data point equal
to 1. Use Euclidean distance on the transformed data to rank the
data points.
2.7. BIBLIOGRAPHIC NOTES
2.7
47
Bibliographic Notes
Methods for descriptive data summarization have been studied in the statistics literature long before the onset of computers. Good summaries of statistical descriptive data mining methods include Freedman, Pisani and Purves
[FPP07], and Devore [Dev95]. For statistics-based visualization of data using
boxplots, quantile plots, quantile-quantile plots, scatter plots, and loess curves,
see Cleveland [Cle93].
Pioneering work on data visualization techniques is described in The Visual
Display of Quantitative Information [Tuf83], Envisioning Information [Tuf90],
and Visual Explanations : Images and Quantities, Evidence and Narrative
[Tuf97], all by Tufte, in addition to Graphics and Graphic Information Processing by Bertin [Ber81], Visualizing Data by Cleveland [Cle93], and Information Visualization in Data Mining and Knowledge Discovery edited by
Fayyad, Grinstein, and Wierse [FGW01]. Major conferences and symposiums
on visualization include ACM Human Factors in Computing Systems (CHI),
Visualization, and the International Symposium on Information Visualization.
Research on visualization is also published in Transactions on Visualization
and Computer Graphics, Journal of Computational and Graphical Statistics,
and IEEE Computer Graphics and Applications.
There have been many graphical user interfaces and visualization tools developed for data mining, which can be found in various data mining products.
Several books on data mining, such as Data Mining Solutions by Westphal and
Blaxton [WB98], present many good examples and visual snapshots. For a survey of visualization techniques, see “Visual techniques for exploring databases”
by Keim [Kei97].
Similarity and distance measures among various variables have been introduced in many textbooks that study cluster analysis, including Hartigan
[Har75], Jain and Dubes [JD88], Kaufman and Rousseeuw [KR90], and Arabie, Hubert, and De Sorte [AHS96]. Methods for combining attributes of
different types into a single dissimilarity matrix were introduced by Kaufman
and Rousseeuw [KR90].
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CHAPTER 2. GETTING TO KNOW YOUR DATA
Bibliography
[AHS96] P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scientific, 1996.
[Ber81] J. Bertin. Graphics and Graphic Information Processing. Berlin,
1981.
[Cle93] W. Cleveland. Visualizing Data. Hobart Press, 1993.
[Dev95] J. L. Devore. Probability and Statistics for Engineering and the Science (4th ed.). Duxbury Press, 1995.
[FGW01] U. Fayyad, G. Grinstein, and A. Wierse. Information Visualization
in Data Mining and Knowledge Discovery. Morgan Kaufmann, 2001.
[FPP07] D. Freedman, R. Pisani, and R. Purves. Statistics (4th ed.). W. W.
Norton & Co., 2007.
[Har75] J. A. Hartigan. Clustering Algorithms. John Wiley & Sons, 1975.
[JD88] A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Prentice Hall, 1988.
[Kei97] D. A. Keim. Visual techniques for exploring databases. In Tutorial Notes, 3rd Int. Conf. Knowledge Discovery and Data Mining
(KDD’97), Newport Beach, CA, Aug. 1997.
[KR90] L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: An
Introduction to Cluster Analysis. John Wiley & Sons, 1990.
[Tuf83] E. R. Tufte. The Visual Display of Quantitative Information. Graphics Press, 1983.
[Tuf90] E. R. Tufte. Envisioning Information. Graphics Press, 1990.
[Tuf97] E. R. Tufte. Visual Explanations : Images and Quantities, Evidence
and Narrative. Graphics Press, 1997.
[WB98] C. Westphal and T. Blaxton. Data Mining Solutions: Methods and
Tools for Solving Real-World Problems. John Wiley & Sons, 1998.
49
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